What is simplify-ln(x)+1/2 ln(x^2+1) ?

The solution to simplify-ln(x)+1/2 ln(x^2+1) is ln(((x^2+1)^{1/2})/x)

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simplify-ln(x)+1/2 ln(x^2+1)

simplify

- ln (x) + \frac{1}{2} ln (x^{2} + 1)

ln \frac{( x ^{2} + 1 ) ^{\frac{1}{2}}}{x}

Solution steps

- ln (x) + \frac{1}{2} ln (x^{2} + 1)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{2} ln (x^{2} + 1) = ln ((x^{2} + 1)^{\frac{1}{2}})

= - ln (x) + ln ((x^{2} + 1)^{\frac{1}{2}})

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln ((x^{2} + 1)^{\frac{1}{2}}) - ln (x) = ln \frac{( x ^{2} + 1 ) ^{\frac{1}{2}}}{x}

= ln \frac{( x ^{2} + 1 ) ^{\frac{1}{2}}}{x}

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What is simplify-ln(x)+1/2 ln(x^2+1) ?
The solution to simplify-ln(x)+1/2 ln(x^2+1) is ln(((x^2+1)^{1/2})/x)